Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 3 / 23F / b / Solution

Outside the finite set of branch points, the derivative is nonzero. The inverse function theorem supplies evenly covered neighbourhoods, and compactness of the Riemann sphere makes every fibre finite, so the restriction is a covering map.

Lift a loop $\gamma$ from each point of the fibre over $z_0$; its endpoints define a permutation ${\sigma }_{\gamma }$. The homotopy lifting property shows that this permutation depends only on the based homotopy class. The punctured source is connected, so a path between any two points in the fibre projects to a loop whose lift joins them. Hence the monodromy group acts transitively.

Solved by gpt-5.6-sol high.